tgEDMD: Approximation of the Kolmogorov Operator in Tensor Train Format

TL;DR

This paper introduces tgEDMD, a tensor train-based method for efficiently approximating the Kolmogorov generator from data, enabling detailed analysis of stochastic dynamical systems with improved data efficiency.

Contribution

The paper presents a novel tensor train approach for approximating the Kolmogorov generator, extending data-driven Koopman analysis to infinitesimal generators with theoretical and practical insights.

Findings

Demonstrates the method's effectiveness on benchmark examples

Provides analysis of consistency and complexity

Shows improved data efficiency in generator approximation

Abstract

Extracting information about dynamical systems from models learned off simulation data has become an increasingly important research topic in the natural and engineering sciences. Modeling the Koopman operator semigroup has played a central role in this context. As the approximation quality of any such model critically depends on the basis set, recent work has focused on deriving data-efficient representations of the Koopman operator in low-rank tensor formats, enabling the use of powerful model classes while avoiding over-fitting. On the other hand, detailed information about the system at hand can be extracted from models for the infinitesimal generator, also called Kolmogorov backward operator for stochastic differential equations. In this work, we present a data-driven method to efficiently approximate the generator using the tensor train (TT) format. The centerpiece of the method is a TT representation of the tensor of generator evaluations at all data sites. We analyze consistency and complexity of the method, present extensions to practically relevant settings, and demonstrate its applicability to benchmark numerical examples.